Phase transitions in growing groups: How cohesion can persist

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Phase transitions in growing groups: How cohesion can persist

Enrico Maria Fenoaltea, Fanyuan Meng, Run-Ran Liu, and Matúš Medo

Phys. Rev. Research 5, 013023 – Published 19 January 2023

Abstract

The cohesion of a social group is the group's tendency to remain united. It has important implications for the stability and survival of social organizations, such as political parties, research teams, or online groups. Empirical studies suggest that cohesion is affected by both the admission process of new members and the group size. Yet, a theoretical understanding of their interplay is still lacking. To this end, we propose a model where a group grows by a noisy admission process of new members who can be of two different types. Cohesion is defined in this framework as the fraction of members of the same type and the noise in the admission process represents the level of randomness in the evaluation of new candidates. The model can reproduce the empirically reported decrease of cohesion with the group size. When the admission of new candidates involves the decision of only one group member, the group growth causes a loss of cohesion even for infinitesimal levels of noise. However, when admissions require a consensus of several group members, there is a critical noise level below which the growing group remains cohesive. The nature of the transition between the cohesive and noncohesive phases depends on the model parameters and forms a rich structure reminiscent of critical phenomena in ferromagnetic materials.

Received 31 August 2022
Accepted 23 December 2022

DOI:https://doi.org/10.1103/PhysRevResearch.5.013023

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Complex systems Continuous phase transition Discontinuous phase transition Social systems

Interdisciplinary Physics

Authors & Affiliations

Enrico Maria Fenoaltea ^1,2,*, Fanyuan Meng ^3,†, Run-Ran Liu ³, and Matúš Medo ^4,2

¹School of Science, Hainan University, Haikou, 570228, People's Republic of China
²Physics Department, University of Fribourg, Chemin du Musée 3, 1700 Fribourg, Switzerland
³Research Center for Complexity Sciences, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China
⁴Department of Radiation Oncology, Inselspital, University Hospital of Bern, and University of Bern, 3010 Bern, Switzerland

^*enrico.fenoaltea@unifr.ch
^†fanyuan.meng@hotmail.com

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Issue

Vol. 5, Iss. 1 — January - March 2023

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Figure 1
Analytical and numerical results when fit and unfit candidates are equally likely. (a) Mean cohesion, $\bar{C}$ , for various ways (UC, PA, DS) of choosing one evaluating member vs. evaluation noise $η$ ( $N_{0} = 1$ and $N = 10^{3}$ ). (b) Scaling of $\bar{C} - 1 / 2$ with group size $N$ for various ways (UC, PA, DS) of choosing one evaluating member ( $N_{0} = 1$ and $η = 0.1$ ). In panels (a) and (b), the solid lines represent the analytical solutions given by Eqs. (3), (7), and (11). The symbols and their error bars (too small to be visible) represent means and standard errors based on 10 000 model realizations. (c) Mean cohesion, $\bar{C}$ , in the UC case with $m = 2$ evaluating members vs evaluation noise $η$ ( $N_{0} = m$ and $N = 10^{3}, 10^{6}, 10^{9}$ ). The solid line is the analytical limit when $N \to \infty$ , given by Eq. (15). (d) The phase diagram of mean cohesion, $\bar{C}$ , in the $(m, η)$ plane for the UC case ( $N = 10^{6}$ and $N_{0} = m$ ). The solid line is the analytical solution in the $N \to \infty$ limit, given by Eq. (17). (e) Scaling of $\bar{C} - 1 / 2$ in the UC case with $m = 4$ evaluating members close to the critical noise level $η_{c} = 3 / 8$ ( $N_{0} = m$ and $N = 10^{3}, 10^{6}, 10^{9}$ ). The solid line is the analytical limit when $N \to \infty$ . (f) A phase diagram in the $(η, N)$ plane comparing multiple evaluators with a dictator. Each point is colored by the smallest number of evaluators that are needed to outperform a single dictator (up to $m = 4$ ). In panels (c)–(f), each point is an average over 500 model realizations (error bars representing twice the SEM are too small to be visible).
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Figure 2
Analytical and numerical results when fit and unfit candidates are not equally likely ( $f \neq 1 / 2$ ). (a) Mean cohesion, $\bar{C}$ , for various ways (UC, PA, DS) of choosing one evaluating member vs evaluation noise $η$ ( $N_{0} = 1$ , $N = 10^{3}$ , and $f = 1 / 4$ ). (b) Mean cohesion, $\bar{C}$ , in the UC case with $m = 2$ evaluating members vs evaluation noise $η$ ( $N_{0} = m$ , $N = 10^{9}$ , and $f = 1 / 4, 1 / 2, 3 / 4$ ). (c) The phase diagram of mean cohesion, $\bar{C}$ , in the $(m, η)$ plane for the UC case ( $f = 1 / 4$ , $N_{0} = m$ , and $N = 10^{6}$ ). (d) Mean cohesion, $\bar{C}$ , in the UC case with $m = 2$ evaluating members vs fraction of fit candidates $f$ ( $η = 0.1 < η_{c} = 1 / 4$ , $N_{0} = m$ , and $N = 10^{3}, 10^{6}, 10^{9}$ ). (e) The phase diagram of mean cohesion, $\bar{C}$ , in the $(f, η)$ plane for the UC case ( $N = 10^{6}$ and $N_{0} = m = 2$ ). (f) A phase diagram in the $(f, η)$ space, comparing multiple evaluators with a dictator. Each point is colored by the smallest number of evaluators that are needed to outperform a single dictator (up to $m = 4$ ). In all panels, each data point is an average over 500 model realizations (error bars representing twice the SEM are mostly too small to be visible). The solid lines represent numerical solutions of Eq. (18) which is valid for $N \to \infty$ .
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